Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}-5x+6y &= -4 \\ -2x+2y &= -3\end{align*}$
Solution: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-1$ and the bottom equation by $3$ $\begin{align*}5x-6y &= 4\\ -6x+6y &= -9\end{align*}$ Add the top and bottom equations. $-x = -5$ Divide both sides by $-1$ and reduce as necessary. $x = 5$ Substitute $5$ for $x$ in the top equation. $-5( 5)+6y = -4$ $-25+6y = -4$ $6y = 21$ $y = \dfrac{7}{2}$ The solution is $\enspace x = 5, \enspace y = \dfrac{7}{2}$.